How to prove a trajectory equation is bounded

Question

I have a trajectory equation:

$$0.0908\log{y}-0.0011 y = -0.0988\log{x}+0.001x+c$$

I wanted to prove that this trajectory is bounded because I need to prove the existence of limit cycles in a prey-predator model.

Answer 1

Consider the equation $$ A \log{x} - B x + C \log{y} - D y = c $$ on $\{(x,y) : x > 0, y > 0\}$, where $A, B, C, D > 0$ and $c \in \mathbb{R}$.

The continuous function $x \mapsto A \log{x} - B x$ has limit $-\infty$ as $x \to 0^{+}$ and as $x \to \infty$, hence is bounded from above. Denote its largest value by $M$. Similarly, the continuous function $y \mapsto C \log{y} - D y$ has limit $-\infty$ as $y \to 0^{+}$ and as $y \to \infty$, hence is bounded from above. Denote its largest value by $N$

Further, there is $\xi > 0$ such that $$ A \log{x} - B x < c - N \quad \text{ for all } x > \xi, $$ and there is $\eta > 0$ such that $$ C \log{y} - D y < c - M \quad \text{ for all } y > \eta. $$

For $(x, y)$ with $x > \xi$ we have $A \log{x} - B x + C \log{y} - D y < (c-N) + N = c$. Similarly, for $(x, y)$ with $y > \eta$ we have $A \log{x} - B x + C \log{y} - D y < M + (c-M) = c$.

We have proved that the set $$ \{(x,y) : A \log{x} - B x + C \log{y} - D y = c \} $$ must be contained in $(0, \xi] \times (0, \eta]$, hence is bounded.